The family of Sarah Murnaghan, the 10-year-old girl whose quest for a lung transplant prompted a change in national transplant policy, said Thursday her first biopsy shows no rejection of her donor lungs.

Sarah, who was born with cystic fibrosis, had the lung transplant June 12 and is still unable to talk because of tubes in her throat. But in a video shared by her family Thursday -- the first one since her surgery -- Sarah waves and mouths "hi" and "I love you" to her mother, sitting off camera.

The family said Sarah had surgery Wednesday to fix her diaphragm to support her new lungs. The biopsy was taken during the operation, they said.

"Sarah had a nice, stable night," the family wrote on a Facebook page set up for Sarah. "Her pain is increased, but they seem to be able to manage it. Overall she seems relieved the surgery is over, sitting waiting yesterday made her very anxious."

The diaphragm surgery was Sarah's fourth in less than a month. Shortly after her lung transplant June 12, the donated lungs failed, and Sarah had emergency surgery to put her on a bypass machine that took over the work of the lungs.

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Her family explained later that the first set of lungs was in poor condition when Sarah received them, but they decided to proceed with the surgery because "Sarah was out of time to wait."

The second set of lungs was infected with pneumonia, but again the family went ahead with the transplant because "they were Sarah's best and only hope." Since then, the family says, her lungs have improved with each day and continue to work better and better.

Sarah was on the transplant list for children's lungs for 18 months. She wasn't able to qualify for adult lungs because she was too young; the agency that oversees transplants, the Organ Procurement and Transplant Network, had a policy that children only 12 or older could be prioritized for adult organs.

The family petitioned Health and Human Services Secretary Kathleen Sebelius to change the rules so Sarah could qualify for adult lungs. They argued the rule was unfair, and several lawmakers also asked Sebelius to act.

On June 5, the Murnaghans asked a federal judge to issue a restraining order to block Sebelius from having the organ network apply the rule. The judge granted the injunction and ordered Sebelius to direct the network to waive the rule in Sarah's case.

The family said they never asked that Sarah be placed in front of anyone on the transplant list who was in a more serious condition.

Days later, the organ network approved a change to their policy to allow lung transplant programs to request additional priority for younger transplant candidates. The change is effective for a year, when it must be approved by the board of directors.